Graph parameters from symplectic group invariants

Graph parameters from symplectic group invariants Guus Regts∗

Bart Sevenster

†

arXiv:1602.02026v1 [math.CO] 5 Feb 2016

February 8, 2016

Abstract In this paper we introduce, and characterize, a class of graph parameters obtained from tensor invariants of the symplectic group. These parameters are similar to partition functions of vertex models, as introduced by de la Harpe and Jones, [P. de la Harpe, V.F.R. Jones, Graph invariants related to statistical mechanical models: examples and problems, Journal of Combinatorial Theory, Series B 57 (1993) 207–227]. Yet they give a completely different class of graph invariants. We moreover show that certain evaluations of the cycle partition polynomial give examples of graph parameters that can be obtained this way. Keywords. partition function, graph parameter, symplectic group, cycle partition polynomial M.S.C [2010] Primary 05C45, 15A72; Secondary 05C25, 05C31

1 Introduction Partition functions of statistical (spin and vertex) models as graph parameters were introduced by de la Harpe and Jones in [9]. Partition functions of spin models include the number of graph homomorphisms into a fixed graph, and they play an important role in the theory of graph limits, cf. [10]. A standard example of the partition function of a vertex model is the number of matchings. Szegedy [18, 19] showed that the partition function of any spin model can be realized as the partition function of a vertex model. Partition functions of vertex models occur in several different mathematical disciplines. For example as Lie algebra weight systems in the theory of Vassiliev knot invariants cf. [4], as tensor network contractions in quantum information theory [11] and as Holant problems in theoretical computer science [2, 3, 21]. In [18], Szegedy calls a vertex model an edge-coloring model and we will adopt his terminology here. Let N include 0 and for k ∈ N we denote by [k] the set {1, . . . , k}. Following Szegedy [18], for k ∈ N and a field F we will call h = (hn ), with hn a symmetric tensor in (F k )⊗n for each n ∈ N, a k-color edge-coloring model over F (in [9] it is called a vertex model). We will often omit the reference to k or the field F. The partition function of h is the graph parameter ph defined for a graph G = (V, E) by ph ( G ) : =

∑ ∏ h( φ:E →[k] v∈V

∗ University

O

e φ ( a) ),

of Amsterdam, Sciencepark 105-107, 1098 XH Amsterdam, the Netherlands. [email protected]. Supported by a NWO Veni grant. † University of Amsterdam. Email: [email protected]

1

(1)

a∈δ( v)

Email:

where for v ∈ V, δ(v) denotes the set of edges incident with v and where e1 , . . . , ek is a basis N N for F k and h( a∈δ(v) eφ( a) ) denotes the coefficient of hdeg(v) at the tensor a∈δ(v) eφ( a) ∈ (Fk )⊗ deg(v) . (Note that since hdeg(v) is symmetric the order is irrelevant.) Szegedy [18] characterized which graph parameters are partition functions of edgecoloring models over R. In [5] and [16] characterizations over algebraically closed fields of characteristic zero are given. These characterizations make use of the invariant theory of the orthogonal group and revealed an intimate connection between these partition functions and this branch of invariant theory. From that perspective it is natural to ask what connections there are for other (matrix) groups. For example, the general linear group is connected to partition functions for directed graphs, cf. [5]. In this paper we will focus on the symplectic group and introduce, and characterize, a class of graph parameters related to tensors invariants of the symplectic group that we call skew-partition functions of edge-coloring models. It turns out that these skew-partition functions are most naturally defined for directed graphs, but, surprisingly, we show that when restricted to skew-symmetric tensors, one can in fact define them for undirected Eulerian graphs. These skew-partition functions are related to ‘negative dimensional’ tensors; see [13] in which Penrose already describes a basic example. For suitable choice of tensors, these skew-partition functions give rise to evaluations of the cycle-partition polynomial (a normalization of the Martin polynomial [12]) at negative even integers. As such, these skew-partition functions play a similar role for the cycle partition polynomial as the number of homomorphisms into the complete graph for the chromatic polynomial. Besides their connection to the symplectic group, the introduction of skew-partition functions is also motivated by a paper of Schrijver [16]. In [16] Schrijver characterized partition functions of edge-coloring models in terms of rank growth of edge-connection matrices. We need some definitions to state this result. For k ∈ N, a k-fragment is a graph which has k vertices of degree one labeled 1, 2, . . . , k. We will refer to an edge incident with a labeled vertex as an open end. Let Fk denote the collection of all k-fragments, where we allow multiple edges and loops. Then F0 can be considered as the collection of all graphs G . Moreover, any component of the underlying graph of a fragment in Fk may consist of a single edge with an empty vertex set; we call this graph the circle and denote it by . Define a gluing operation ∗ : Fk × Fk → G , where for two k-fragments F1 , F2 , F1 ∗ F2 is the graph obtained from F1 ∪ F2 by removing the labeled vertices and gluing the edges incident to equally labeled together. Note that by gluing two open ends of which the endpoints are labeled one creates a circle. For a graph parameter f : G → C define the k-th connection matrix M f ,k , by M f ,k ( F1 , F2 ) := f ( F1 ∗ F2 ) for F1 , F2 ∈ Fk . Schrijver [16] proved f : G → C is the partition function of an edge-coloring model if and only if f (∅) = 1, f ( ) ∈ R and rk( M f ,k ) ≤ f ( )k for all k ∈ N.

(2)

Since the rank of a matrix can never be negative the conditions in (2) implicitly imply that f ( ) ≥ 0. It is however natural to ask what happens when we keep the rank conditions only for even k and add the condition that f ( ) < 0. Squaring a parameter satisfying these conditions, we automatically obtain the partition function of an edge-coloring model, as follows from (2). There are graph parameters that satisfy these conditions. Consider for 2

example f (G) =



(−2)c( G) if G is 2-regular, 0 otherwise,

(3)

where c( G ) denotes the number of components of the graph G. Setting f ( ) = −2, one has rk( M f ,2k ) ≤ 4k . (This follows for example from our main theorem, cf. Theorem 1.) We note that no matter what value we choose for , this f is never the (ordinary) partition function of an edge-coloring model, as follows from [5], cf. [14, Proposition 5.6]. It turns out that skew-partition functions are graph parameters satisfying the modified conditions in (2), and moreover, these are the only graph parameters. This is the content of our main theorem. The paper is organized as follows. In the next section we introduce skew-partition functions, derive some basic properties and state our main theorem, cf. Theorem 1. In Section 3 we show that evaluations at negative even integers of the cycle partition polynomial can be realized as the skew-partition function of suitable edge-coloring models. Section 4 deals with some framework, which is needed to prove Theorem 1. This framework is similar to the framework developed in [5]. In particular, the connection with the invariant theory of the symplectic group will become clear there. In Section 5 we prove Theorem 1 and in Section 6 we conclude with some further remarks and some questions.

2 Definitions and main result In this section we introduce skew-partition functions and state our main result. In what follows all graphs are allowed to have loops and multiple edges. We call a graph Eulerian if each vertex has even degree. In particular, the circle is an Eulerian graph.

2.1 Definitions and main result Let G = (V, E) be an Eulerian graph with an Eulerian orientation ω of E. Definition 1. A local ordering κ = (κ − , κ + ) of the edges at each vertex v is compatible with ω if κv− is a bijection between {1, 3, ..., d(v) − 1} and δ− (v), the incoming arcs at v and κv+ is a bijection between {2, 4, ..., d(v)} and δ+ (v), the outgoing arcs at v. A local ordering κ that is compatible with ω decomposes E uniquely into circuits of the form (v1 , e1 , ..., ei , vi , ei+1 , ..., v1 ), where κv−i (ei ) + 1 = κv+i (ei+1 ) for each i. (Recall that a circuit is a closed walk where vertices may occur multiple times, but edges may not). We will refer to a circuit in this decomposition as a κ-circuit. Let c( G, κ ) be the number of κ-circuits in this decomposition. Let G # be the set of triples ( G, ω, κ ), where G is an Eulerian graph, ω is an Eulerian orientation of G and κ is a local ordering of the edges at each vertex of G which is compatible with ω. Let Vℓ := C2ℓ and let e1 , . . . , e2ℓ be the standard basis of Vℓ . Define for i = 1, . . . , 2ℓ f i ∈ Vℓ by  −ei+ℓ if i ≤ ℓ, (4) f i := ei−ℓ if i > ℓ. We denote by n (Vℓ ) the space of skew-symmetric n-tensors; note that this space is zero V L 2ℓ V n V if n > 2ℓ. Let (Vℓ ) := (Vℓ ). We view elements of (Vℓ ) as linear functions n =0 V

3

⊗n n =V 0 Vℓ

L 2ℓ

h∈

→ C. For an even positive integer d we set O(d) := {1, 3, . . . , d − 1}. For (Vℓ ) and ( G, ω, κ ) ∈ G # we now define sh (( G, ω, κ )) = (−1)c( G,κ )

∑

∏

φ:E ( G )→[2ℓ] v∈V ( G )

h(

O

eφ(κv (i)) ⊗ f φ(κv (i+1)) ).

(5)

i∈O ( d( v))

Proposition 1. Let G be an Eulerian graph. Then the value of sh (( G, ω, κ )) for ( G, ω, κ ) ∈ G # is independent of the choice of Eulerian orientation ω and compatible local ordering κ. We postpone the proof of this proposition to the end of this section. It allows us to define the following graph parameter. Definition 2. Let h ∈ invariant defined by

V

(Vℓ ). Then we define the skew-partition function1 of h to be the graph

sh ( G ) : =



sh (( G, ω, κ )) 0

for any ( G, ω, κ ) ∈ G # , if G is Eulerian, otherwise.

(6)

Notice that sh ( ) = −2ℓ. Let us give an example. Example 1. Let ℓ = 1 and let h = −e1 ⊗ e2 + e2 ⊗ e1 . Then  (−2)# components of G if G is 2-regular, sh ( G ) = 0 otherwise. This example can be generalised quite a bit. Readers familiar with the cycle partition polynomial will recognise that sh ( G ) is equal to the cycle partition polynomial evaluated at −2. In Section 3 we shall show that we can realize evaluations of the cycle partition polynomial at any negative even integer as the partition function of a suitable skew-symmetric tensor. One of our main results in this paper is a characterization of graph parameters that are skew-partition functions of edge-coloring models, which we will state below after we introduce some terminology. A graph parameter f : G → C is called multiplicative if f ( F1 ∪ F2 ) = f ( F1 ) f ( F2 ) for all F1 , F2 ∈ G and f (∅) = 1. For a graph G = (V, E) and U ⊆ E of size ℓ + 1, label the vertices incident with U with the numbers [2ℓ + 2] by first arbitrarily directing the edges in U and the labeling the starting points with [ℓ + 1] and the endpoints with [2ℓ + 2] \ [ℓ + 1] (some vertices may receive multiple labels this way). Then for π ∈ S2ℓ+2 , let GU,π be the graph obtained from G by removing U from E and adding the edges {π (i), π (ℓ + 1 + i)} for i ∈ [ℓ + 1] to E \ U (any labeling would do, but for future purposes this labeling is most convenient). We can now state our main theorem, characterizing skewpartition functions, which we prove in Section 5. Theorem 1. Let f : G → C be a graph parameter. Then the following are equivalent: (i) there exists a skew-symmetric tensor h ∈

V

(Vℓ ) for some ℓ ∈ N such that f ( G ) = sh ( G ),

(ii) f is multiplicative, f (∅) = 1, f ( G ) = 0 if G is not Eulerian, and for some ℓ ∈ N, f ( ) = −2ℓ and ∑π ∈S2ℓ+2 f ( GU,π ) = 0 for each graph G = (V, E) ∈ E and U ⊆ E of size ℓ + 1, (iii) f (∅) = 1, f ( ) < 0 and for each k ∈ N, the rank of M f ,2k is bounded by ( f ( ))2k . Moreover, the equivalence between (i) and (ii) holds with the same ℓ. 1 The

name skew-partition function is chosen so as to distinguish it from the ordinary partition function; it is moreover motivated by the fact that sh ( G ) can be seen as the contraction of a tensor network with respect to a skew-symmetric form.

4

2.2 Proof of proposition 1 Before proving the proposition, we introduce a quantity that will turn out to be useful later V on. For ( G, ω, κ ) ∈ G # , h ∈ (Vℓ ) and a map φ : E( G ) → [l ] consider sh,φ ( G, ω, κ ) := (−1)c( G,κ )

∑

∏

h(

ψ:E ( G )→{0,ℓ} v∈V ( G )

O

e(φ+ψ)(κv (i)) ⊗ f (φ+ψ)(κv (i+1)) ),

(7)

i∈O ( d( v))

where (φ + ψ) : E( G ) → [2ℓ] is defined as e 7→ φ(e) + ψ(e) for e ∈ E( G ). Lemma 2. Let ( G, ω, κ ) ∈ G # and let φ : E( G ) → [ℓ]. Let C be a κ-circuit and let ω ′ be the Eulerian orientation obtained from ω by inverting the orientation of the edges in C. Let κ ′ be obtained from κ by flipping the values of two consecutive edges of C incident with a vertex v for all vertices v of C. Then sh,φ ( G, ω, κ ) = sh,φ ( G, ω ′ , κ ′ ). Proof. Write G = (V, E). We may assume that no component of G is a circle. Define for ψ : E → {0, ℓ}, o(ψ) := |ψ−1 (0)|, the number of edges that get assigned 0 by ψ. We can now, by definition of the f i , rewrite (7) as follows:

(−1)c( G,κ )

∑

(−1)o(ψ)

ψ:E →{0,ℓ}

∏

O

h(

v ∈V ( G )

e(φ+ψ)(κv (i)) ⊗ e(φ+ψ)(κv (i+1))+ℓ ),

(8)

i∈O ( d( v))

where the addition is carried out modulo 2ℓ. Let for ψ : E → {0, ℓ} ψ′ : E → {0, ℓ} be defined by  ψ( e) if e 6∈ C, ψ′ ( e) : = ψ(e) + ℓ mod 2ℓ if e ∈ C.

Write H = ( G, ω, κ ) and H ′ = ( G, ω ′ , κ ′ ). We now compare the contribution of ψ in sh,φ ( H ) and ψ′ in sh,φ ( H ′ ). Clearly, o(ψ) and o(ψ′ ) differ by |C |. At any vertex v of C with incoming edge e and outgoing edge e′ , we see e(φ+ψ)(e) ⊗ e(φ+ψ)(e′)+ℓ in sh,φ ( H ), while in sh,φ ( H ′ ) we see e(φ+ψ)(e′)+2ℓ ⊗ e(φ+ψ)(e)+2ℓ. Since h is skew-symmetric these contributions differ by a minus sign. So by (8) we conclude that ψ and ψ′ have the same contribution. This proves the lemma. We now conclude this section by proving the following lemma, which clearly implies Proposition 1. Lemma 3. Let ( G, ω, κ ) ∈ G # and let h ∈ (Vℓ ). Then for any map φ : E( G ) → [ℓ], the value sh,φ ( G, ω, κ ) is independent of the choice of ω and κ. V

Proof. Suppose we have ( G, ω, κ ) and ( G, ω, κ ′ ) in G # . If we apply a transposition to κv+ or κv− at some v, then the parity of the number of circuits changes. As h is skew-symmetric, the evaluation of the tensor at v cf. (8) also changes sign, so these cancel out. So we can apply permutations at each v to κ to go from κ to κ ′ without changing the value of sh,φ . Now consider ( G, ω, κ ) and ( G, ω ′ , κ ′ ). The symmetric difference of ω and ω ′ , i.e., the set of edges where they do not give the same orientation is again a Eulerian graph with Eulerian orientation given by the restriction of ω. Let C be a κ-circuit in the symmetric difference of ω and ω ′ . Let ω ′′ be obtained from ω by inverting the orientation of the edges of C and let κ ′′ be obtained from κ by flipping the values of two consecutive edges of C incident with a vertex v of C for each vertex v of C. By Lemma 2 this does not change the value of sh,φ . Repeating this until there are no circuits left in the symmetric difference finishes the proof. 5

3 Evaluations of the cycle partition polynomial The cycle partition polynomial, first introduced, in a slightly different form, by Martin in his thesis [12], is related to Eulerian walks in graphs and to the Tutte polynomial of planar graphs. Several identities for the cycle partition polynomial were established by Bollobás [1] and Ellis-Monaghan [6]. For a graph G = (V, E) let C( G ) be the collections of all partitions of E into circuits. For C ∈ C( G ), let |C | be the number of circuits in the partition. The cycle partition polynomial J ( G, x) is defined as J ( G, x) := ∑ x|C| . C∈C( G )

So if G is not an Eulerian graph, then J ( G, x) = 0. We clearly have that J ( G ∪ H, x) = J ( G, x) J ( H, x) and J ( , x) = x. For n ∈ N, we can express J ( G, n) as n

J ( G, n) =

∑ ∏ ∏(degA (v) − 1)!!, i

A n ( G ) v ∈V i = 1

(9)

where An ( G ) ranges over ordered partitions of E into n subsets A1 , ..., An such that (V, Ai ) is Eulerian for all i, and where for a positive odd integer m, m!! := m(m − 2) · · · 1, cf. [1, 6]. To express (9) as the partition function of an n-color edge-coloring model, let hk ∈ (C n )⊗k N be defined by taking for φ : [k] → [n] the value ∏ki=1 (φ−1 (|{i }| − 1)!! at i∈[k] eφ(i) , where we set for convenience (−1)!! = 1 and m!! = 0 if m is even. Then ph ( G ) = J ( G, n) for each graph G. We will similarly show that evaluations of J ( G, x) at negative even integers can be realized as skew-partition functions of skew-symmetric tensors. It follows from work of Bollobás [1] (see also [6]) that the evaluation of the cycle partition polynomial J ( G, x) of a graph at negative even integers can be expressed as J ( G, −2ℓ) =

ℓ

(−2)∑i=1 | Hi | ,

∑

(10)

H1 ,...,Hℓ

where this sum runs over all ordered partitions of the edge set of G into 2-regular subgraphs H1 , . . . , Hℓ and | Hi | denotes the number of components of the graph induced by Hi . Proposition 4. For every ℓ ∈ N, there is an h ∈ graph G.

V

(Vℓ ) such that sh ( G ) = J ( G, −2ℓ) for each

be the collection of all subsets of [ℓ] of size k. Let h be the Proof. For k = 0, . . . , ℓ, let ([ℓ] k) skew-symmetric tensor given by ℓ

h :=

∑ ∑ ∑

sgn(π )(−1)k

k=0 T ∈([ℓ]) π ∈S T

O

eπ (t) ⊗ eπ (t+ℓ),

(11)

t∈ T

k

where ST is the group of permutations on T ∪ ( T + ℓ). We will show that J ( G, −2ℓ) = sh ( G ). Take any ( G, ω, κ ) ∈ G # . Let φ : E( G ) → [ℓ] and let for i ∈ [l ], Hi := φ−1 (i ). Abusing notation, the graph induced by Hi will be denoted by Hi . We will show that the contribution of sh,φ ( G, ω, κ ) to (7) is equal to ( 0 if some Hi is nonempty and not 2-regular, (12) ℓ (−2)∑i=1 | Hi | if each Hi is either 2-regular or empty. 6

By (10) this implies the proposition. To prove (12), suppose first that some nonempty Hi is not 2-regular. Then Hi has a vertex v of odd degree. Fix any ψ : E( G ) → {0, ℓ} and consider φ′ = φ + ψ. If deg(v) = 1, we cannot see both color i and i + ℓ at v implying by (8) that the contribution of φ′ to the skew-partition function is zero. Similarly, if deg(v) > 2, we see color i or i + 1 at least twice at v implying by skew-symmetry of h that the contribution of φ′ is zero. Suppose now that each Hi is either empty or 2-regular. Let us choose another Eulerian orientation ω ′ by choosing a cyclic orientation of all cycles of each of the Hi . Take a compatible local ordering κ ′ which is in addition compatible with the cycles. Then c( G, κ ′ ) = ∑ℓi=1 | Hi |. By Lemma 3 we know that sh,φ ( G, ω, κ ) = sh,φ ( G, ω ′ , κ ′ ). For ψ : E( G ) → {0, ℓ}, the contribution of ψ to the sum (7) is zero if ψ is not constant on each cycle, by skewsymmetry of h. So let us assume that ψ is constant on each cycle. Consider first the case that ψ(e) = 0 for all e ∈ E( G ). Then o(ψ) = | E( G )| and so the parity of o(ψ) is equal the parity of the number of vertices with degree 2 mod 4. Thus by definition of h, the ℓ contribution of ψ is equal to (−1)∑i=1 | Hi | . We next show that if ψ′ is obtained from ψ by changing the value on a cycle C of some Hi , then they have the same contribution. This follows from the fact that the parities of o(ψ) and o(ψ′ ) differ by the parity of |C | and the fact that at each vertex of C we have interchanged ei ⊗ ei+ℓ with ei+ℓ ⊗ ei resulting in a factor of (−1)|C| by the skew-symmetry of h. By (8), this proves (12) and finishes the proof of the proposition. We remark that the cycle partition polynomial evaluated at x ∈ / Z cannot be realized as the partition function, nor as the skew-partition function, of any edge-coloring model. This follows from Theorem 1 and (2) and the fact that by [16, Proposition 2] the rank of the submatrix of M J (·,x ),2k indexed by fragments in which each component consists of an unlabeled vertex with two open ends incident with it, has rank at least k!. The case that x is a negative odd integer is discussed in Section 6.

4 Framework In this section we will look at ordered directed graphs, i.e., directed graphs in which each vertex is equipped with a total order of the arcs incident with it. We will define skewpartition functions for these ordered directed graphs and characterize a class of these functions based on the Nulstellensatz and the invariant theory of the symplectic group, similar to the characterization of partition functions of edge-coloring models in [5]. This characterization will then be used to prove Theorem 1 in Section 5.

4.1 Ordered directed graphs and skew-partition functions An ordered directed graph is a triple (V, A, κ ), where (V, A) is a directed graph (we allow multiple arcs and loops), and where κ = (κv )v∈V with κv : [deg(v)] → δ(v) a bijection for v ∈ V; giving δ(v), the arcs incident with v, a total ordering at v (here deg(v) denotes the total degree of v). We note that in case a is a directed loop at v, a occurs twice in δ(v). Let G ∗ denote the collection of ordered directed graphs. We often just write graph instead of ordered directed graph for elements of G ∗ . We call a map f : G ∗ → C an ordered directed graph parameter if f is constant on isomorphism classes of ordered directed graphs.

7

For G = (V, A, κ ) ∈ G ∗ and φ : A → [2ℓ] (for some ℓ ∈ N) we set o(φ) := |φ−1 ([ℓ])|, the number of images of φ that land in [ℓ]. For v ∈ V we let φv be the map φv : [deg(v)] → [2ℓ] defined as follows: for i ∈ [deg(v)], φv (i ) = φ(κv (i )) φv (i ) = φ(κv (i )) + ℓ

mod 2ℓ

if κv (i ) ∈ δ− (v), if κv (i ) ∈ δ+ (v),

(13)

where δ+ (v) is the set of outgoing arcs and δ− (v) is the set of incoming arcs at v. Let Vℓ := C2ℓ for some ℓ ∈ N and let h = (hn )n∈N with hn ∈ Vℓ⊗n ; we call h a 2ℓ-color edgecoloring model. (We often omit the reference to ℓ.) The skew-partition function of h is the ordered directed graph parameter sh : G ∗ → C defined for G = (V, A, κ ) ∈ G ∗ by

∑

sh ( G ) =

(−1)o(φ)

∏ h ( φv ) ,

(14)

v ∈V

φ:A →[2ℓ]

where h(φv ) is equal to the value of hdeg(v) at eφv := eφv (1) ⊗ · · · ⊗ eφv (deg(v)) . (Here e1 , . . . , e2ℓ is the standard basis of C2ℓ .) The name skew-partition function is motivated by the fact that if G ′ is obtained from G by flipping the direction of an arc of G then sh ( G ′ ) = −sh ( G ), as we will see later, but which is also not difficult to show directly. We can view a triple ( G, ω, κ ) ∈ G # , of which no component of G is a circle, as an V ordered directed graph G ′ . In terms of the skew-partition we have for h ∈ (Vℓ ) that sh (( G, ω, κ )) = (−1)c( G,κ ) sh ( G ′ ).

4.2 Invariants of the symplectic group Let h·, ·i be the nondegenerate skew-symmetric bilinear form on Vℓ := C2ℓ given by h x, yi =   0 I T 2 ℓ , with I the ℓ × ℓ identity matrix. x Jy for x, y ∈ C , where J is the 2ℓ × 2ℓ matrix −I 0 Note that h·, ·i naturally induces a nondegenerate symmetric bilinear form on (Vℓ )⊗2m for any m, which again will be denoted by h·, ·i. Let e1 , . . . , e2ℓ be the standard basis for Vℓ and let f1 , . . . , f2ℓ be the associated dual basis with respect to the skew-symmetric form, i.e., h f i , e j i = δi,j for all i, j = 1, . . . , 2ℓ. Then V f i is defined by (4). A basis for n (Vℓ ) is given by {eS | S ⊆ [2ℓ], |S| = n}, where for S = {s1 , . . . , sn }, with s1 < s2 < · · · < sn , eS : =

∑

sgn(π )esπ (1) ⊗ · · · ⊗ esπ (n) .

π ∈ Sn

Let R = O( (Vℓ )∗ ) be the ring of regular functions of the space of skew-symmetric tensors. V Let for S ⊂ [2ℓ], yS ∈ (Vℓ )∗ be defined by yS (eS′ ) = δS,S′ for S′ ⊂ [2ℓ]. Then R can be identified with the ring of polynomials in the variables yS with S ⊆ [2ℓ], The symplectic group Spℓ is the group of 2ℓ × 2ℓ matrices that preserve the skew-symmetric form; i.e., for g ∈ C2ℓ×2ℓ , g ∈ Spℓ if and only if h gx, gyi = h x, yi for all x, y ∈ Vℓ . Note V that the symplectic group has a natural action on R; for g ∈ Spℓ , p ∈ R and v ∈ (Vℓ ), ( gp)(v) := p( g−1 v). For a set X we denote by CX the vectorspace of finite formal C-linear combinations of elements of X. We will define a linear map p : C G ∗ → R that will turn out to have as image the space of Spℓ -invariant polynomials. To do so we need a definition. For an injective map V

8

φ : [k] → [2ℓ] we define the sign of φ to be the sign of the permutation π ∈ Sk such that for ψ = φ ◦ π we have ψ(1) < ψ(2) < · · · < ψ(k). This is denoted by sgn(φ). Define for φ : [k] → [2ℓ] elements of R by zφ : =



sgn(φ)yφ([k]) if φ is injective, 0 otherwise.

(15)

We now define the linear map p : C G ∗ → R by G = (V, A, κ ) 7→

∑

(−1)o(φ)

φ:A →[2ℓ]

∏ zφ

v

(16)

v ∈V

for G ∈ G ∗ . Note that if we evaluate p( G ) at h ∈ (Vℓ ) we just get the partition function of h as defined in (14), since zφv (h) = h(φv ). So p( G )(h) = sh ( G ). To describe the kernel of the map p we need some definitions. For G = (V, A, κ ) ∈ G ∗ and U ⊆ A of size ℓ + 1 identify U with [ℓ + 1] and let for π ∈ S2ℓ+2 Aπ be the set of arcs obtained from A by first removing the arcs in U from A and secondly by labeling for i ∈ [ℓ + 1] the starting vertex of arc i with i and its terminating vertex with ℓ + 1 + i (some vertices may obtain multiple labels this way) and by then adding (π (i ), π (ℓ + 1 + i )) to A \ U for each arc i ∈ [ℓ + 1]. Let GU,π = (V, Aπ , κ ). Let Jℓ ⊂ C G ∗ be the subspace spanned by V

{

∑

sgn(π ) GU,π | G = (V, A, κ ) ∈ G ∗ , U ⊆ A, |U | = ℓ + 1}.

(17)

π ∈S2ℓ+2

Note that while GU,π may depend on the choice of identification of U with [ℓ + 1], the sum ∑π ∈S2ℓ+2 sgn(π ) GU,π does not. So the space Jℓ is well defined. Call G ′ a negative flip of G if G ′ is obtained from G by flipping the direction of an odd number of its arcs. The group ∏v∈V Sdeg(v) acts on ordered directed graphs with the degree sequence ∏ v∈V deg(v) by setting π · (V, A, κ ) = (V, A, κ ◦ π ) for π ∈ ∏v∈V Sdeg(v) and an ordered directed graph G. Call G ′ a negative permutation of G if G ′ = π · G for an odd permutation π. Let I ⊂ C G ∗ be the subspace spanned by

{G + G ′ , G + G ′′ | G ∈ G ∗ , G ′ a negative flip of G, G ′′ a negative permutation of G } Let finally

I ℓ = Jℓ + I .

(18)

Proposition 5. The image of p is equal to RSpℓ , the space of Spℓ -invariant polynomials in R, and the kernel of p is equal to Iℓ . We postpone the proof of this proposition to the next section. First we will utilize it to characterize which ordered directed graph parameters are partition functions of skewsymmetric tensors. Theorem 2. An ordered directed graph parameter f : G ∗ → C is the partition function of a skewV symmetric tensor h ∈ (Vℓ ) if and only if f is multiplicative and the linear extension of f to C G ∗ is such that f (Iℓ ) = 0.

9

Proof. Skew-partition functions are clearly multiplicative. By Proposition 5, we have that V for any h ∈ (Vℓ ) that sh (Iℓ ) = 0, proving the ‘only if’ part. The proof of the ‘if’ direction is based on a beautiful, and by now, well-known idea of Szegedy, cf. [18]; see also [5]. We will sketch the proof. The idea is to use the Nullstellensatz V to find a solution h ∈ (Vℓ ) to the set of equations f ( G ) = p( G )(h) with G ∈ G ∗ . Since f is multiplicative and maps Iℓ , the kernel of p, to zero there is a unique algebra homoV morphism fˆ : RSpℓ → C such that f = fˆ ◦ p. If there is no solution h ∈ (Vℓ ) to the set of equations f ( G ) = p( G )(h) with G ∈ G ∗ , then by Hilbert’s Nullstellensatz, 1 is contained in the ideal generated by f ( G ) − p( G ). In other words, there exist ordered directed graphs G1 , . . . , Gn and r1 , . . . , rn ∈ R such that n

1=

∑ ri ( f (Gi ) − p(Gi )).

(19)

i=1

As the image of p is equal to RSpℓ , applying the Reynolds operator of the symplectic group (i.e., the projection R 7→ RSpℓ onto the space of the Spℓ -invariants) to both sides of (19), we may assume that each ri is equal to p(ηi ) for some linear combination of ordered directed graphs ηi . Now applying fˆ to both sides of (19) we obtain 1=

n

n

i=1

i=1

∑ fˆ( p( Hi ))( f (Gi ) − fˆ( p(Gi ))) = ∑ fˆ( p( Hi ))( f (Gi ) − f (Gi )) = 0,

a contradiction. This finishes the proof. Remark 1. We would like to remark here that Theorem 2 remains valid if we replace skewsymmetric tensors by say symmetric tensors. We just need to appropriately adjust the ideal I , but other than that all arguments remain valid. The proof of Proposition 5 in the next section can also be easily adapted to this setting. Remark 2. We have not used the circle in our characterization, but if we want to allow the circle as an ordered directed graph, then for a 2ℓ-color edge-coloring model h we should define sh ( ) = −2ℓ to be compatible with our characterization. Following up on this, in [17] Schrijver characterizes weight systems f coming from a representation of a Lie algebra (i.e., functions on certain 4-regular ordered directed graphs with two incoming and two outgoing arcs at each vertex that satisfy a certain relation) in terms of rank growth of associated connection matrices. We note here that the condition that f ( ) ≥ 0 should be added to statement (iv) of his theorem, since skew-partition functions can give examples of weight systems that satisfy the rank growth, but have f ( ) < 0.

4.3 Proof of Proposition 5 To prove Proposition 5 we will make use of the first fundamental theorem of invariant theory for the symplectic group and a result of Hanlon and Wales [8]. For a graph G = (V, A, κ ) ∈ G ∗ , let n := |V |, identify V with [n] and let D = ( D1 , . . . , Dn ) be its degree sequence; i.e., Di = deg(i). For a degree sequence D we let GD∗ be the set of graphs with degree sequence D and we let R D be the space of polynomials where each monomial is equal to ∏ ni=1 ySi with Si ⊆ [2ℓ] and |Si | = Di . Let 2m := ∑ni=1 Di . ∗ . To prove Proposition 5 it suffices to show that Write pD for the restriction of p to C G D ∗ im pD = ( R D )Spℓ and ker pD = C G D ∩ Iℓ .

10

(20)

− → To show (20), let Mm denote the collection of directed perfect matchings on [2m]. We will next define maps τ, σD and µ D so as to make the following diagram commute: ∗ CGD

pD

µD

− → C Mm

RD σD

τ

Vℓ⊗2m .

(21)

− → Let τ : C Mm → Vℓ⊗2m be the unique linear map defined by M 7→

∑

2m O

M aφ,j

(22)

φ: [2m ]→[2ℓ] j=1 |φ( e )|=1 for all e ∈ E ( M )

− → M is equal to e for M ∈ Mm , where aφ,j φ( j) if j is the tail of an arc of M, and equal to f φ( j) if j is the head of an arc of M. Note that since the tensor ∑2i=ℓ 1 ei ⊗ f i is skew-symmetric, it follows that if M ′ is obtained from M by flipping c of its arcs, then τ ( M ′ ) = (−1)c τ ( M ).

(23)

To define the map µ D , let P = { P1 , . . . , Pn } be the partition of the set [2m] where P1 := {1, . . . , D1 }, P2 := { D1 + 1, . . . , D1 + D2 }, etc. Then µ D is the unique linear map defined by sending a directed matching M to the graph G obtained from M by identifying the vertices in each class of P . For each vertex v of G the bijection κv : [deg(v)] → δ(v) is obtained from the (natural) ordering of [2m]. The map σD is the unique linear map defined by sending eφ(1) ⊗ · · · ⊗ eφ(2m) for φ : [2m] → [2ℓ] to the monomial ∏ni=1 zφi where φi : [ Di ] → [2ℓ] is defined as φi ( j) := φ( j + D1 + · · · + Di−1 ), and where zφi is defined in (15). − → Writing out the definition of σD (τ ( M )) for M ∈ Mm , it is clear that it is equal to pD (µ D ( M )). So the diagram (21) commutes. In particular, since µ D is surjective, it implies by (23) that if G ′ is obtained from G by flipping the direction of an arc, then p( G ) = − p( G ′ ). − → The symmetric group S2m has a natural action Vℓ⊗2m . It also acts naturally on Mm ; − → − → for π ∈ S2m and M ∈ Mm we define πM ∈ Mm to be the directed matching with arcs (π (i), π ( j)) for (i, j) ∈ E( M ). Let us denote by Sn ⊆ S2m the subgroup that permutes the sets P1 , . . . , Pn , but maintains the relative order of the Pi . Then Sn , as a subgroup of S2m , has − → a natural action on C Mm and V ⊗2m . − → Let SD := ∏ ni=1 SPi ⊆ S2m . We let SD act on C Mm and V ⊗2m with the sign representation; − → i.e., for π ∈ SD , M ∈ C Mm , and v ∈ V ⊗2m we set π · M := sgn(π )πM and π · v := sgn(π )πv. Then τ is Sn and SD -equivariant. By the first fundamental theorem for the symplectic group, cf. [7, Section 5.3.2], we have that − → (24) τ (C Mm ) = (Vℓ⊗2m )Spℓ . Sp

Let now q ∈ R D ℓ . Then, as σD is surjective, there exists v ∈ Vℓ⊗2m such that σD (v) = q. Since σD is Spℓ -equivariant (since V ∼ = V ∗ as Spℓ -modules), we may assume that v is Spℓ -invariant. 11

− → So by (24) there exists M ∈ C Mm such that τD ( M ) = v. Then by the commutativity of (21) it follows that pD (µ D ( M )) = q, showing ⊇ of the first part of (20). As the inclusion ⊆ is trivial, we have the first part of (20). ∗ . Then, since µ is surjective, we know Suppose now that pD (γ) = 0 for some γ ∈ C G D D − → there exists M ∈ C Mm such that µ D ( M ) = γ. Let StabSn ( D ) := {π ∈ Sn | πD = D }. Since ∗ are unlabeled), γ is invariant under the action of Sn (as the vertex sets of the graphs in G D we may assume that M is StabSn ( D )-invariant. Let v := τ ( M ). Then, as τ is Sn -equivariant, v is StabSn ( D )-invariant. By the commutativity of (21) we have that σD (v) = 0. This implies that v is contained in the space spanned by {u − π · u | u ∈ Vℓ⊗2m , π ∈ SD , sgn(π ) = −1}. − → − → As C Mm / ker τ ∼ = τ (C Mm ), and τ is SD -equivariant we find that M = M1 + M2 with M2 ∈ ker τ and M1 contained in the space spanned by − → {u − π · u | u ∈ C Mm , π ∈ SD , sgn(π ) = −1}. Clearly, µ D ( M1 ) is contained in Iℓ . We will now describe the kernel of τ to show that µ D ( M2 ) is contained in Iℓ as well. Consider two directed matchings M, N. Then their union is a collection of cycles. Fix for each cycle an orientation and call an arc a ∈ E( M ) ∪ E( N ) odd if it is traversed in the opposite direction. The number of odd arcs in E( M ) ∪ E( N ) is denoted by o( M ∪ N ). Note that the parity of the number of odd arcs is independent of the choice of orientation of the cycles, as they are all of even length. Let us fix M0 to be the directed matching with arcs (1, 2), (3, 4), . . . , (2m − 1, 2m) and define (25) v0 := ∑ sgn(ρ)ρM0 , ρ∈S2ℓ+2

where we consider S2ℓ+2 as a subgroup of S2m acting on [2ℓ + 2] ⊆ [2m]. We call a matching N a negative flip of a matching M if it is obtained from M by flipping the directions of an odd number of arcs of M. Lemma 6. The kernel of the map τ is spanned by

− → {πv0 , M + M ′ | π ∈ S2m , M ∈ Mm , M ′ a negative flip of M }. Before proving this lemma, let us first observe that it implies that the kernel of τ is ∗ . This implies that γ ∈ I , showing that ker p ⊆ I. It is conversely mapped onto Iℓ ∩ C G D D ℓ ∗ ⊆ ker p . straightforward using (21) to see that Iℓ ∩ C G D D So we are done by giving a proof of Lemma 6. Proof of Lemma 6. To prove the lemma we will use a result of Hanlon and Wales [8], which deals with matrices indexed by undirected matchings. We will start by relating the kernel of the map τ to the kernel of an associated matrix. − → We define two matrices, A and B indexed by M2m as follows: A M,N := (−2ℓ)c( M∪ N )

and

B M,N := (2ℓ)c( M∪ N ) (−1)o( M∪ N )+m ,

12

− → for M, N ∈ M2m , where c( M ∪ N ) denotes the number of cycles in M ∪ N. Note that both A and B are S2m -invariant. We will now show: the kernel of τ is equal to the kernel of B.

(26)

To prove (26), we first note that the bilinear form h·, ·i restricted to the image of τ is nondegenerate. Indeed, let v ∈ (Vℓ⊗2m )Spℓ be nonzero. Then we know that there exists u ∈ Vℓ⊗2m such that hv, ui 6= 0. Since for each g ∈ Spℓ we have hv, gui = h g−1 v, ui = hv, ui, we see that, by integrating hv, gui over a maximal compact subgroup of Spℓ with respect to the Haar measure (that is, by applying the Reynolds operator), we may assume that u is Spℓ -invariant. In other words, there exists u ∈ (Vℓ⊗2m )Spℓ such that hv, ui 6= 0, as desired. − → Now fix M, N ∈ Mm . We will show that hτ ( M ), τ ( N )i = B M,N . Fix for each cycle in M ∪ N an orientation. Flip all odd arcs in M ∪ N to obtain directed matchings M ′ and N ′ respectively. By (23), we have hτ ( M ), τ ( N )i = (−1)o( M∪ N ) hτ ( M ′ ), τ ( N ′ )i. Now

hτ ( M ′ ), τ ( N ′ )i =

∑

2m

′

′

N M i. , aψ,j ∏haφ,j

(27)

j=1 φ,ψ: [2m ]→[2ℓ] |φ( e )|=1 for all e ∈ E ( M ′ ) |ψ( e ′ )|=1 for all e ′ ∈ E ( N ′ )

Since M ′ ∪ N ′ does not contain any odd arcs we see that ( heφ( j) , f ψ( j) i if j is the tail of an arc of M ′ M′ N ′ h aφ,j , aψ,j i = h f φ( j) , eψ( j) i if j is the head of an arc of M ′ .

(28)

For each cycle of C of M ′ ∪ N ′ we see that, to get a nonzero contribution, we need for each j ∈ V (C ) that ψ( j) = φ( j). So each cycle C gives a contribution of (−1)|V (C)| 2ℓ. Hence ′ ′ hτ ( M ′ ), τ ( N ′ )i = (−1)m (2ℓ)c( M ∪ N ) . We conclude that hτ ( M ), τ ( N )i = B M,N , as desired. Now since the form h·, ·i is nondegenerate on the image of τ, we find that the kernel of B is indeed equal to the kernel of τ, proving (26). It follows from Hanlon and Wales [8, Theorem 3.1] that the kernel of A is spanned by the set − → { ∑ πρM0 , M − M ′ | π ∈ S2m , M ∈ Mm , M ′ a flip of M }. (29) ρ∈S2ℓ+2

We will now show how to relate the kernel of A to the kernel of B. Define for a matching M, sgn( M ) to be the sign of any permutation π such that πM = M0 . This is well defined since the stabilizer of M0 in S2m consists of permutations of even − → sign. First we show that for any M, N ∈ Mm we have:

(−1)o( M∪ N ) = sgn( M )sgn( N )(−1)c( M∪ N ) .

(30)

To see this we may assume that the underlying graph of M is equal to the underlying graph of M0 (by applying the same permutation to both M and N). Choose an orientation of the cycles in M ∪ N. Multiplying both sides of (30) by (−1)o( M∪ N ) , we may assume that o( M ∪ N ) = 0. Then for each cycle C in M ∪ N we need to apply a cycle πC ∈ S2m to N in order to map N to M. Since these cycles are of even length and pairwise disjoint, it follows that the product of their signs is equal to (−1)c( M∪ N ) . Hence sgn( N ) = (−1)c( M∪ N ) sgn( M ), proving (30). 13

− → This implies for any µ : Mm → C:

∑−→

∑−→

µ M M ∈ ker A ⇐⇒

M ∈ Mm

sgn( M )µ M M ∈ ker B.

(31)

M ∈ Mm

Indeed, for any fixed matching N we have:

(−1)m ( Bµˆ ) N =

∑−→

sgn( M )(−1)m µ M BN,M = sgn( N )

M ∈ Mm

∑−→

µ M A N,M .

M∈M m

Now combining (29) with (31) we obtain the Lemma.

5 Proof of Theorem 1 5.1 The equivalence (i) ⇔ (ii) We will derive the equivalence of (i) and (ii) from Theorem 2. Let E ⊂ G be the collection of Eulerian graphs, where we do not allow circles and let E ∗ ⊂ G ∗ be the collection of ordered directed graphs with all degrees even. Let H = (V, E) ∈ E and ω be an Eulerian orientation of H and let κ be a compatible local ordering. When considering ( H, ω, κ ) as an ordered directed graph, we will refer to it as ι ω,κ ( H ). Let now ω ′ be another Eulerian orientation of H, with compatible local ordering κ ′ . Then by the proof of Proposition 1 we have that ′ (−1)c( H,κ ) ι ω,κ ( H ) = (−1)c( H,κ ) ι ω ′ ,κ ′ ( H ) mod I . This implies that we have a unique map ι : E → C G ∗ /I

given by

H 7→ (−1)c( H,ω,κ ) ι ω,κ ( H )

for any choice of Eulerian orientation ω and compatible local ordering κ. It is clear that extending ι linearly to C E we obtain a linear bijection ι : C E → C E ∗ /(I ∩ C E ∗ ). So the composition p ◦ ι : C E → R is a well-defined map. Let Iℓ ⊂ C E be the linear space spanned by (32) { ∑ HU,π | H = (V, E) ∈ E , U ⊆ E, |U | = ℓ + 1}. π ∈S2ℓ+2

We will now show ker( p ◦ ι) = Iℓ .

(33)

Since any ordered directed graph G ∈ E ∗ is congruent to ι ω,κ ( H ) or −ι ω,κ ( H ) modulo the ideal I for some ( H, ω, κ ) ∈ G # , and since ι : C E → C E ∗ /(I ∩ C E ∗ ) is a bijection, in order to prove (33), it suffices to show that for any G = (V, A, κ ) ∈ E ∗ and U ⊆ A of size ℓ + 1 and underlying graph H = (V, E) ∈ E , we have

∑

ι( HU,π ) = ±

π ∈S2ℓ+2

∑

sgn(π ) GU,π

mod I .

(34)

π ∈S2ℓ+2

Since ι( H ) = ± G mod I , we may assume that there exists an Eulerian orientation ω for H and a compatible local ordering κ, such that ι ω,κ ( H ) = G. To compute HU,π for π ∈ S2ℓ+2 it is convenient to give the endpoints of the edges in U the same labels as in G (the sum ∑π ∈S2ℓ+2 HU,π does not depend on the chosen direction of the edges of U). We will show ι( HU,π ) = sgn(π )(V, Aπ , κ )

14

mod I .

(35)

This clearly implies (34). It suffices to show (35) for the case that π is equal to the transposition (i, j) for i, j ∈ [2ℓ + 2] (as transpositions generate the symmetric group). The Eulerian orientation ω and compatible local ordering κ in H naturally induce a Eulerian orientation ω ′ and compatible local ordering κ ′ in HU,π . Then it is easy to see that ′ (−1)c( H,κ ) = (−1)c( HU,π ,κ )+1 as either two circuits in H are transformed into a single circuit in HU,π or conversely a circuit in H is transformed into two circuits in HU,π ; see Figure 1. So this proves (35) and we conclude that we have (33). P1

P1

u

a1

j

u

v

i

a2′

j

a2

i P2

a1′

v

P2

Figure 1: The transposition (i, j) Now the equivalence between (i) and (ii) follows directly from Theorem 2. Indeed, let f : G → C be such that f ( G ) = 0 if G is not Eulerian, and extend f linearly to C G . Then we obtain a unique linear map fˆ : C G ∗ → C such that fˆ(ι( H )) = f ( H ) for all H ∈ E , fˆ( H ) = 0 if H ∈ G ∗ contains a vertex of odd (total) degree and such that fˆ(I) = 0. If f = sh for some V h ∈ (Vℓ ), then by (33) we have that f ( Iℓ ) = 0, proving that (i) implies (ii). Conversely, V if f satisfies (ii), then by (34) we have that fˆ(Iℓ ) = 0. So there exists h ∈ (Vℓ ) such that fˆ is the skew-partition function of h. By definition we have that f ( G ) = sh ( G ) for G ∈ G , finishing the proof.

5.2 The implication (iii) ⇒ (ii) Notice that, as the rank of M f ,0 = 1 and f (∅) = 1, f is multiplicative. Using the theory developed by Hanlon and Wales [8], we first show that f ( ) has to be even. Let N f ,2k be the submatrix of M f ,2k indexed by the matchings on 2k vertices, i.e., the entries of N f ,2k are of the form f ( )n for some n ∈ N. So N f ,2k depends only on f ( ). Schrijver [16, Proposition 3] shows that if rk( N f ,2k ) ≤ f ( )2k for all k ∈ N, then f ( ) ∈ Z. Notice that if f ( ) ∈ N, then rk( N f ,2k ) ≤ f ( )2k . In Section 3 of [8] it is explained that the eigenspaces of N f ,2k are in one-to-one correspondence with partitions of 2k and that their dimension can be computed with the hook-length formula. If f ( ) = n, then we denote N f ,2k by Nn,2k . Lemma 7. For n ∈ N, we have that rank( N−n,2k ) ≤ n2k for all k ∈ N if and only if n is even. Proof. If n = −2, then the partition (2, ..., 2) of 2k is the only partition that corresponds to an eigenspace with non-zero eigenvalue of N−2,2k cf. [8, Theorem 3.1]. By the hook-length

15

formula cf. [15], we have that rank( N−2,2k ) =

(2k)! ≤ 22k . (k + 1)!k!

The matrix N−2n,2k is the Schur-product of N−2,2k and Nn,2k , so rk( N−2n,2k ) ≤ (2n)2k . If n = 2m − 1, then for even d the partitions λd = (2m + d, 2m, ..., 2m) of 2md + d give eigenspaces with non-zero eigenvalue of Nn,2md+d , cf. [8, Theorem 3,1]. Denote the dimension of the corresponding eigenspace with f d . Following [16] we have, with the hook-length formula that (2md + d)! fd = , (d!)2m+1 p(d) where p(d) is a polynomial in d. So by Stirling’s formula

( f d )1/(2md+d) → 2m + 1, as d → ∞, showing that the rank of M−n,k is larger than (n + 1)2k for k large enough. Lemma 7 implies that f ( ) = −2ℓ for some ℓ ∈ N. To prove the remaining part of the implication we use some framework developed by Schrijver [16], which we will now describe. We can equip C F2k with an associative multiplication by letting F1 F2 be the 2kfragment obtained from the disjoint union of F1 and F2 by deleting the vertex labeled k + i of F1 and the vertex labeled i of F2 and gluing the two edges incident to these vertices together for i = 1, . . . , k and extending this bilinearly to C F2k . This makes C F2k into an associative algebra and the unit 1k in C F2k is given by k disjoint labeled edges such that the ends are labeled i and i + k. Let I2k := {γ ∈ C F2k | f (γ ∗ F ) = 0 for all F ∈ F2k }. Then I2k is an ideal and Ak := C F2k /I2k is an associative algebra. Define τ : Ak → C for x ∈ Ak by τ ( x ) = f ( x ∗ 1k ). Using this function τ, one can show that the algebra Ak is semisimple [16]. Futhermore,  τ ( x) is a negative integer if k is odd, (36) if x is a non-zero idempotent in Ak , then τ ( x) is a positive integer if k is even. This is essentially Proposition 6 in [16]; the proof is exactly the same. Let k, m ∈ N, then, following [16], for π ∈ Sm , let Pk,π be the 2km-fragment consisting of km disjoint edges ei,j for i = 1, ..., m and j = 1, ..., k, where ei,j connects the vertices labeled j + (i − 1)k and km + j + (π (i ) − 1)k. We define qk,m to be qk,m :=

∑

Pk,π .

π ∈ Sm

Let o(π ) be the number of orbits of the permutation π. If m > (2ℓ)k and k is odd, we have that τ (qk,m ) =

∑

((−2ℓ)k )o(π ) = (−1)m

π ∈ Sm

= (−1)

∑

(−1)m−o(π ) (2ℓ)ko(π )

π ∈ Sm m

∑

sgn(π )(2ℓ)

π ∈ Sm

16

ko ( π )

= 0,

(37)

since ∑π ∈Sm sgn(π ) xo(π ) = x( x + 1) · · · ( x − m + 1). Using this, we first show that f satisfies

∑

f ( GU,π ) = 0 for each graph G = (V, E) ∈ E and U ⊆ E of size ℓ + 1.

(38)

π ∈S2ℓ+2 1 Let m = 2ℓ + 2 and consider q1,m . Then by (37) τ (q1,m ) = 0. Since m! q1,m is an idempotent, (36) implies that q1,m = 0 in Aℓ+1 . In other words, q1,m ∈ Im . Consider a graph G = (V, E) and let U ⊆ E of size ℓ + 1. Let F ∈ F2k be obtained from G as follows: replace each edge e = {u, v} from U by two open ends; one connected to u and the other to v. Then, as q1,m ∈ Im , 0 = f ( F ∗ q1,m ) = ∑ f ( GU,π ), π ∈ Sm

proving (38). To show that f ( G ) = 0 if G is non-Eulerian, fix a vertex v of G that has odd degree, say k. Define fragments F0 , F1 ∈ Fk as follows: F0 consists of a single vertex with k open ends incident with this vertex; F1 is obtained from G by removing v from G and replacing each edge incident with v with an open end. Then F0 ∗ F1 = G. Now take m such that 1 qk,m is an idempotent and by (37), τ (qk,m ) = 0 and so by (36), qk,m is m > (2ℓ)k . Then m! actually 0 in Akm . Now, take m copies of both F0 and F1 and create a fragment F ∈ Fkm from their disjoint union as follows: the end labeled j in Fi gets label ikm + j + k(n − 1) in the n-th copy of Fi . Then, as qk,m ∈ Imk , 0 = f ( F ∗ qk,m ) = m!( f ( F0 ∗ F1 )m ) = m!( f ( G )m ). So f ( G ) = 0, finishing the proof.

5.3 The implication (i) ⇒ (iii) Fix k ∈ N. For an Eulerian 2k-fragment (i.e., all unlabeled vertices have even degree), an Eulerian orientation is defined to be an orientation of the edges such that for each unlabeled vertex the number of incoming arcs equals the number of outgoing arcs and such that there are exactly k labeled vertices incident with an incoming arc (note that such an orientation always exists, which can for example be see by temporarily adding an edge between the vertices labeled i and k + i for i = 1, . . . , k). # be the set of triples ( F, ω, κ ), where F is an Eulerian 2k-fragment, none of whose Let F2k components are circles, with ω an Eulerian orientation and κ a local ordering compatible with ω at all non-labeled vertices. Notice that, similar to what we have seen earlier, the local ordering κ decomposes E( F ) into circuits and k directed walks that begin and end at a labeled vertex. Denote by c( F, κ ) the number of circuits in this partition and let M = M ( F, κ ) be the directed perfect matching on [2k], where (i, j) ∈ E( M ) if and only if there is a directed walk from i to j in this partition. # ), a map ψ : [2k] → [2ℓ] and a map φ : E ( F ) → [2ℓ] we say that φ For ( F, ω, κ ∈ F2k extends ψ if for each i ∈ [2k] and the unique open end e ∈ E( F ) incident with the vertex labeled i we have that φ(e) = ψ(i ). By o′ (φ) we denote the number of edges, that are not open ends, that get mapped into [ℓ] by φ. Now define the tensor th ( F, ω, κ ) ∈ Vℓ⊗2k as follows:   O ′ (−1)c( F,κ ) eφ( i) ⊗ f φ( j) . (39) ∑ (−1)o (φ) ∏ h(φv ) ∑ ψ: [2k]→[2ℓ]

φ:E ( F )→[2ℓ] φ extends ψ

v ∈V ( F )

17

( i,j)∈ E ( M )

# and let M = M ( F , κ ) for i = 1, 2. Fix Let ( F1 , ω1 , κ1 ) and ( F2 , ω2 , κ2 ) be elements of F2k i i i an arbitrary orientation of the cycles in M1 ∪ M2 and consider the odd arcs of M1 ∪ M2 . For i = 1, 2 and an odd arc ( a, b) ∈ E( Mi ) change the orientation of ωi by flipping the directions of the edges in the path from a to b in the partition of E( Fi ) and change the local ordering at each vertex of the path by interchanging the edges in the path incident with the vertex. Call the resulting orientations and local orderings ωi′ and κi′ respectively, and note that κi′ is compatible with ωi′ . Now ω1′ and ω2′ induce an Eulerian orientation ω in F1 ∗ F2 . Moreover, κ1′ and κ2′ induce a compatible local ordering κ in F1 ∗ F2 . By definition we have

sh ( F1 ∗ F2 ) = (−1)k+c( M1 ∪ M2 ) hth ( F1 , ω1′ , κ1′ ), th ( F2 , ω2′ , κ2′ )i, where the (−1)k factor is to compensate for the skew-symmetry of the form, cf. (28), and where the (−1)c( M2 ∪ M2 ) factor comes from the circuits formed by gluing F1 and F2 . By the same argument as in the proof of Lemma 2 we have that th ( Fi , ωi , κi ) = (−1)o( Mi ) th ( Fi , ωi′ , κi′ ) for i = 1, 2, where o( Mi ) denotes the number of odd arcs of Mi in M1 ∪ M2 . As, by (30) (−1)c( M1 ∪ M2 ) = (−1)o( M1 ∪ M2 ) sgn( M1 )sgn( M2 ), we conclude that sh ( F1 ∗ F2 ) = sgn( M1 )sgn( M2 )(−1)k hth ( F1 , ω1 , κ1 ), th ( F2 , ω2 , κ2 )i.

(40)

Now (40) implies that we can write Msh ,2k as the Gramm matrix (with respect to h·, ·, i) of vectors in Vℓ⊗2k , implying that its rank is bounded by (2ℓ)2k .

6 Concluding remarks and questions Future work Having introduced and characterized skew-partition functions for graphs, the story does not end here. Consider for m ∈ Z the graph parameter f m defined by  c( G) m if G is 2-regular, (41) f m ( G ) := 0 otherwise. Where c( G ) is the number of components of G. Then if m ∈ N we have that f m is the (ordinary) partition function of an edge-coloring model; if m ∈ −2N, then f m is the skewpartition function of a skew-symmetric tensor. This leaves open the case that m is negative and odd. Le us look at the special case that m = −1. Set f −1 ( ) = −1. Then it can be shown, using for example the results of Halon and Wales [8], that rk( M f −1 ,k ) ≤ 3k for each k, as well as that rk( M f −1 ,k ) > 2k for k large enough, cf. Lemma 7. This implies that f −1 is neither a partition function nor a skew-partition function. For a connected graph G we can however realize f −1 ( G ) as the sum f −2 ( G ) + f 1 ( G ), a sum of a skew-partition function and a partition function, and extend this multiplicatively to disjoint unions. The associated models live in C ⊗2 and (C2 )⊗2 respectively. Taking their direct sum we could think of it as living in (C ⊕ C2 )⊗2 . Equipping C with a nondegenerate bilinear form and C2 with a nondegenerate skew-symmetric form we can interpret C as the even component and C2 as the odd component of the super vector space C ⊕ C2 . It is natural to try to define (and characterize) partition functions for models living in a super vector space. In particular we believe that evaluations of the circuit partition polynomial at odd negative integers can realized in this way. This will be the object of future study. Computational complexity For partition functions of edge-coloring models with 2 colors 18

(seen as Holant problems) there is a surprising dichotomy result [3, 21] saying that it is #P hard to evaluate these unless the model satisfies some particular conditions, in which case the partition function can be evaluated in polynomial time. Are similar results true for the skew-partition functions introduced in the present paper? As far as we know even the computational complexity of evaluating the cycle partition polynomial at negative even integers 2ℓ with ℓ < 1 is still open. For the directed cycle partition polynomial this is #P hard even for regular planar graphs of degree 4, except at x = 0 and x = −2, as follows from a reduction to the Tutte polynomial. See [6, 20] for details.

Acknowledgements The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n◦ 339109. We thank Lex Schrijver for useful comments on a earlier version of this paper.

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